What causes cold feet and tingling in feet?

The correct way is ARE your feet cold. Since you are refering to the plural noun, feet :)

Sitting on it for a long time

I am not exactly sure, but i think it is because you lose most of your body heat from your feet, which makes them colder.

John was very eager to do a bungee jump, but when he seen the height he'd have to jump he got "Cold Feet"

Unfortunately, I have nothing scientific to add into this answer, just experience. I have found on a number of occasions that having cold wet feet for a prolonged time can bring on a cold the next...